The Manin-Drinfeld theorem has various equivalent statements. Let $\Gamma$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$. Then:
- for any congruence subgroup $\Gamma$ of $\mathrm{SL}_2(\mathbb{Z})$, the subgroup of $\operatorname{Jac}(X(\Gamma))$ generated by the cusps is finite;
- if $c_1, c_2$ are cusps and $f \in S_2(\Gamma)$ is a Hecke eigenform, the integral $\int_{c_1}^{c_2} f(z) \mathrm{d}z$ is a linear combination of the periods $\Omega_+(f)$ and $\Omega_-(f)$ with coefficients in the field generated by the Hecke eigenvalues of $f$;
- the natural map $H^1_c(Y(\Gamma), \mathbb{Q}) \to H^1(Y(\Gamma), \mathbb{Q})$ has a unique Hecke-equivariant section.
I'm interested in extensions of this to the context of arithmetic quotients of $\mathrm{GL}_2(\mathbb{A}_K)$, where $K$ is a number field. The first statement only makes sense if $K$ is totally real, but the second and third can be formulated for any $K$. Are they true in this generality?
(I have a translation of a 1978 paper by Kurcanov which gives the proof for $K$ an imaginary quadratic field; and I believe there is a more recent paper of Kurcanov that covers CM fields, but I can't read Russian and there doesn't seem to be an English translation.)
